2.1.68 \(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath

problem number 68

Added June 3, 2019.

Problem 3.17(b) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4 \]

Mathematica


\begin {align*} & \left \{u(x,y)\to -\sqrt {-x y-\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y-\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to -\sqrt {-x y+\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y+\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\ \end {align*}

Maple


\[u \left (x , y\right ) = \sqrt {-x y -\sqrt {x^{4}+x^{2} y^{2}+4 \mathit {\_F1} \left (x y \right )}}\]

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